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Full text of "Mathematical And Physical Papers - Iii"

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condition, and therefore steady motion would have been possible. By determining the arbitrary constants, and substituting in % we should have obtained
8     a     r ,     2    fr\l~*l   '   a -5 ---- +r  & -       f sm0,
2    r    a    2  o\aJ    J
Since 8 is supposed to be extremely small, it follows from these expressions that when r is not greater than a moderate multiple of a, the velocities J?, @ are extremely small; but, however small be S, we have only to go far enough from the cylinder in order to find velocities as nearly equal to  Fcos#, + Fsin# as we please. But the distance from the cylinder to which we must proceed in order to find velocities E, @ which do not differ from their limiting values  Fcos#, -f Fsin# by more than certain given quantities, increases indefinitely as 8 decreases. Hence, restoring to the fluid and the cylinder the velocity F, we see that in the neighbourhood of the cylinder the motion of the fluid does not sensibly differ from a motion of translation, the same as that of the cylinder itself, while the distance to which the cylinder exerts a sensible influence in disturbing the motion of the fluid increases indefinitely as S decreases.
48. When we have formed the equations of motion of a fluid on any particular dynamical hypothesis, it becomes a perfectly definite mathematical problem to determine the motion of the fluid when a given solid, initially at rest as well as the fluid, is moved in a given manner, or to discuss the character of the analytical solution in any extreme case proposed. It is quite another thing to enquire how far the principles which furnished the mathematical data of the problem may be modified in extreme cases, or what will be the nature of the actual motion in such cases. Let us regard in this point of view the case considered in the preceding article as a mathematical problem. When the quantity of fluid carried with the cylinder becomes considerable compared with the quantity displaced, it would seem that the motion must become unstable, in the sense in which the motion of a sphere